Fastest way to learn mental math


I need to become faster with mental math for a couple of reasons. Because of that I am looking for a mental math system to get fast results.
I already tried to train with a Soroban. It turned out that I don’t have the time, that it would take, to become proficient.

Don’t get me wrong. I’m not looking for something, that will miraculously boost my mental math skills. I know, that such things don’t exist. I have about 40 minutes every day (when I am walking to work and back), to train my mental math skills but I don’t know, where to start.

Do you have any recommandations for a good comprehensive method?

Thank you.

I answered almost the same question here, 5 months ago:

Did you do anything with that advice? I hope you tried all systems and can now explain which ones work better than others.
If not, start there.

I personally use most systems. What I use most and what is not always written about in books is how to make calculations easier.

If I need to answer 99 X 98, I never calculate that using criss cross for example or other system.
I start with 100 X 100 and work out the difference from there.

A multiplication of 4 digits by 4 digits can have 8 digits at the most. Most people are not capable of working with 8 digits at a time in their head. So start with smaller multiplications.

Find your limit.
Can you do a 2 X 2?
Can you do a 3 X 3 or a 2 X 3?

Let me know, please and we will take it from there.

Train yourself to do at least the following:

  • difference of squares
  • Arthur Benjamin also called the Anchor system
  • criss-cross

Did you at least check those out? Or do you need to still learn those?
I don’t mind if you didn’t. Life gives you other priorities sometimes.
I just need to know where to start and which advice can be skipped.

Some extra tips here:

Also, just start asking question what the best way is for a specific problem you are working on and that needs the be worked out in your head.


And Ywan,

You call this thread ‘Fastest way to learn mental math’.
Imho, there is no fastest way to learn mental math.
It is much better to see it as a life long process with periods where you learn new systems, methods, or short cuts.
And periods where you put these into practice and learn how your mind reacts.

I try to observe what my mind does and learn/improve/optimize form there.
For example I have an auditive way of mental calculation.
I need to hear myself speak numbers in my mind and I need to repeat numbers a couple of times in my mind to store them.
When I was younger I did not know this.
This also limits my speed of calculation. For me, this is ok.
This also means I cannot be as fast as Arthur Benjamin and never will be.
However; I can and do use his methods.

Forget about speed for now. Learn methods and focus on precision and correctness.
When your calculations are correct and you can learn to rely on your mind to do this, then slowly focus on speed.
If you start out with speed but cannot calculate correctly, you cannot build up to a 4X4 which needs correctness and lots of mental sub steps.

I can help you along this process and I need you to give me feedback. The more precise your feedback, the faster you will grow.

Thanks for your reply Kinma.
I tried Trachtenberg and I like the system, but I think, that it is more suited for calculations with pen and paper (because you get the numbers form right to left).
Next I tried the system of Arthur Benjamin. I found it to be good, but I read too much and was confused by all the rules. Because I had not that much time then, I stopped practising.
Now I have more time again and so I want to practice again.

I looked into the criss-cross method, but it didn’t appeal to me. Maybe I need to give it a real try.
The Difference of squares looks interesting, but to use this method, I have to learn to square first. :slight_smile:

2x2 is not a real problem (I remember some of Benjamin’s techniques), although I am not that fast, but the answer is correct. I haven’t tried 2x3 or 3x3, since I stopped reading Benjamin’s book.
I haven’t done any division yet, so I don’t have a clue, how to do it. I haven’t reached this part of Benjamin’s book, when I had to stop.

It would be great, if you could help me. Thank you very much.

Hi Ywan,

Let’s revisit the basic way of this system. If you understand this, you can do most - if not all - of the calculations Art does.

Let’s do 23 X 62.
You can change the ‘23’ into ‘20’, so go down in amount, then the 62 must come up, and becomes 65. So do 20 X 65 = 1300. Easy start, right?
Now the difference.
We subtracted 3 from 23 to get 20. The difference between 20 and 62 is 42.
Now multiply the differences: 3 X 42 = 126.
Add the two results:
1300 + 126 = 1426. This is your answer.

You can also change the ‘23’ into ‘25’, so go up in amount, then the 62 must come down, and becomes 60. So do 25 X 60 = 1500. Easy start, right?
Now the difference.
We added 2 to 23 to get 25. The difference between 25 and 62 is 37.
Now multiply the differences: 2 X 37 = 74.
Subtract the two result numbers:
1500 - 74 = 1426. This is your answer.

Compare the two. Either you go up - with the lowest number - or you go down.

Try this out on a number of multiplications and see if this clicks.
If it does, try to learn both ways (of going either up or down).
Usually either one of these two gets you the easiest numbers to work with.

This is the most generic multiplication system. It work always and will generate digits. If all shortcuts fail, this will generate the digits.
For me it is a last resort way. First I like to find shortcuts. If that does not turn up anything, I use this.

The drawback is that sometimes it leads to a lot of carrying.
Take 99 X 98 for example.

An example of the criss-cross method:
34 X

Start with 10 X 30 = 300
Then do 10 X 4 + 2 X 30 = 40 + 60 = 100. This is the cross.
Add the last two results: 300 + 100 = 400.
Last we do 2 x 4 = 8.
Add 400 and 8 to get 408.

This is true.

Use 2X2 to get reacquainted with the systems.
Then venture to 2X3 and 3x3.

Arthur Benjamins book - if I recall correctly - does not do division.

However; I have written a lot lately about about different ways of speeding this up.
For division, read this (in opposite order):

You said you need to be able to do mental calculation in your work. Can you give examples?

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Hi, I too am a learner of mental math. I mainly just visualize the soroban for any addition or subtraction that I do(and multiple methods for division and multiplication). In my opinion, I’ve gotten very good at it over the years.

I can recommend a system to you, but you first must elaborate on your needs and current abilities. A few to start with:

  1. Can you add 2 three-digit numbers in your mind, provided they are written down?
  2. If so, can you do it without them being written? That is, can you arbitrary choose two large numbers and find their sum without any external aid?
    The same(1 and 2) for subtraction. Please tell your benchmark for each. For example, I can add two 6-digit numbers without any aid. If they are written down, I break them into chunks of 3 digits each and add them. Don’t worry about speed, just make sure you’re completely accurate.

If you can add and subtract pretty well, multiplication and division become relatively easy to learn. An example: Let’s say I need to multiply 249 and 568. This can be simplified to (98) + (2408+5609) + (2456) = 72 + (1920+ 5040) + (134400)
= 141432
As you can see, multiplication requires a heavy deal of addition. Honestly, this is way easier than it looks, so don’t be alarmed by the numbers.

As Kinma said, it would help a lot if elaborated on what your work entails. I’ll need to know if you have a pencil and paper to write down the answer at the very least, or if you would have to do it entirely in your head.

Edit: I noticed that you’ve mentioned one of Arthur Benjamin’s works. I haven’t personally read any of his books, so could you mention which one you’re talking about?

@ Kinma: Thanks for the explanation and the tips on division. I have to take a look at them.
I am going to practice 2x2, till I get the techniques right, then I’m going to move on to 2x3 and 3x3.
You are right, I said, that I need to be able to do mental calculation in my work, but that was a couple of months ago. Currently I don’t need mental calculation for my work. At the moment I just want to learn it for fun and because I think, that it is a good method to train the brain.

@ nkp: You are right, addition and substraction play an important part in mental math.
I want to do everything in my head, without pen and paper.
I read “Secrets of Mental Math” from Benjamin, but I haven’t finished it yet.

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Like I said in the earlier thread:

Don’t settle for one system.
When training, try to do calculations in different ways. So do the calculation 3 to 4 times using a different system at each time. This helps you to see the best shortcut later.
I often ask myself ‘can I find an even better way to calculate this?’.

Train yourself to do at least the following:

  • difference of squares
  • Arthur Benjamin also called the Anchor system
  • criss-cross

If you have to do 21 times 63, why not set this up step by step, like this;
20 times 60 equals 1,200.
Then move to 21 x 60 using the previous result.
‘I say to myself, if 20 times 60 equals 1200, then 21 times 60 equals 1,260.’
Then move to 21 x 63 using 1,260.
‘I say to myself, if 21 times 60 equals 1260, then 21 times 63 equals 1,260 + 3x21.’
Add 3 x21 = 63 to 1260 to get: 1,323.

Last week I had to calculate 62.50 X 34.
One option is to do 62 X 34 and add 17 (0.50 X 34).
However; I decided to do it as a 4X2 (actually a 3X2, since the last digit is zero).
First 62.50 X 30 = 1,875.
Then 62.50 X 4 = 250.
Adding 1,875 and 250 makes: 2,125.

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I know this doesn’t matter but I worked that out in like 4 seconds and I have been slack on mental math recently because it’s hard to find where to start so that was a confidence boost. (instead of doing 100x100 = 10000 then working out the difference I thought 99 is 1 from 100 and 98 is 2 from 100 decided to subtract 2 from 99 and got 97 then multiplied 1 and 2 and thought 02 so my answer was 9702) after a little research I concluded that this was a Vedic maths technique of some form that I happened to remember after months of not using these techniques and doing more anchor/Arthur Benjamin style techniques. I had only used this technique once prior this year in early June I believe and I still managed to remember it randomly. how bizarre